In this paper, we consider the H∞ filtering problem for linear systems using quantized measurements. The communication channel we consider consists of two cases: the ideal one and the unreliable one. For the ideal channel, we designed a filter to mitigate the quantization effects, which ensured not only the asymptotical stability but also a prescribed H∞ filtering performance. For the unreliable channel, we introduced the stochastic variable satisfying Bernoulli random binary distribution to model the lossy measurements. We also designed a filter to cope with the losses and mitigate quantization effects simultaneously which ensured not only stochastic stability, but also a prescribed H∞ filtering performance. Furthermore, we derive sufficient conditions for the existence of the above filters. Finally, a numerical example is given to illustrate that the proposed approach is effective and feasible.